Noncommutative complete intersections and matrix integrals

1. Introduction 1.1. In this paper we work with finitely generated unital associative algebras over C. Such an algebra F is said to be smooth if Ker [F ⊗ F → F ], the kernel of the multiplication map, is a projective F -bimodule. Fix a smooth algebra F , for instance a free algebra F = C x1 , . . . , xd . Let J be a two-sided finitely generated ideal in F. Definition 1.1.1. The algebra A = F/J (or the pair J ⊂ F ) is called a noncommutative complete intersection (NCCI) if J/J 2 is a projective A-bimodule. This is equivalent to the condition that A has a projective A-bimodule resolution of length ≤ 2, cf. Theorem 3.1.1. NCCI algebras have been considered, under various different names, by a number of authors, see e.g. [AH], [An], [GS], [Go], [Pi]. In the present paper, we are interested in the case of graded NCCI algebras. Specifically, fix a finite set I and write R for the algebra of functions I → C, with pointwise multiplication. Let Q be a finite quiver with vertex set I , and assign to each edge a ∈ Q an arbitrary positive grade degree d(a) > 0. This makes the path algebra of Q a free graded R-algebra; moreover, any free graded R-algebra F may be obtained in this way.
Abstract. We introduce a class of noncommutatative algebras called representation complete intersections (RCI). A graded associative algebra A is said to be RCI provided there exist arbitrarily large positive integers n such that the scheme Repn A, of n-dimensional representations of A, is a complete intersection. We discuss examples of RCI algebras, including those arising from quivers. There is another interesting class of associative algebras called noncommutative complete intersections (NCCI). We prove that any graded RCI algebra is NCCI. We also obtain explicit formulas for the Hilbert series of each nonvanishing cyclic and Hochschild homology group of an RCI algebra. The proof involves a noncommutative cyclic Koszul q A, and a matrix integral similar to the one arising in quiver gauge theory. complex, K cyc
arXiv:math/0603272v3 [math.AG] 23 Jan 2007
NONCOMMUTATIVE COMPLETE INTERSECTIONS AND MATRIXGINZBURG
To Bob MacPherson on the occasion of his 60th birthday
Table of Contents
1. 2. 3. 4. 5. 6. Introduction Representation functor and matrix integrals Noncommutative complete intersections The trace map Additional results and examples Preprojective algebras and quiver varieties
1
We study algebras of the form A = F/J, where J is a graded two-sided ideal in the path algebra F such that J/J 2 is a projective finitely generated A-bimodule. Thus, A = r ≥0 A[r ] is a graded algebra such that A[0] = R. The grading on A induces a natural grading on each Hochschild homology group HH k (A). Our main results (Theorem 2.2.7 and Theorem 3.7.7) give explicit formulas for the Hilbert series of each Hochschild homology group of A for a class of algebras A that we call asymptotic representation complete intersections (see below). For k = 0, for instance, Hochschild homology reduces to the commutator quotient space, HH 0 (A) = A/[A, A], where [A, A] denotes the C-linear subspace of A spanned by the commutators ab − ba, a, b ∈ A. The grading on A clearly descends to the commutator quotient. Our result yields a formula, in terms of an infinite product of determinants, for the Hilbert series of the graded algebra Here, A+ = r>0 A[r ], so A+ /[A, A] is the positive degree part of A/[A, A]. We remark that the symmetric algebra in (1.1.2) is equipped with the total grading that comes from the grading on A+ /[A, A] (this grading should not be confused with the standard grading Sym = k≥0 Symk , on the Symmetric algebra of a vector space). 1.2. The main idea of our approach is based on the notion of representation functor and may be outlined as follows. Let A be any graded finitely presented algebra. For each positive dimension d, one has an affine scheme Repd A of d-dimensional representations of A, cf. §2.1. Heuristically, one expects that the collection of representation schemes Repd A provides, as d → ∞, a ‘good approximation’ of the algebra A (this is not true for nongraded algebras which may have no finite dimensional representations at all). In particular, one should be able to read off a lot of information about A from some geometric information about the corresponding representation schemes. In more detail, for each d, the general linear group GLd acts naturally on the scheme Repd A by base change transformations of the underlying vector space of the d-dimensional representation. We consider the algebra C[Repd A]GLd , of GLd -invariant polynomial functions on Repd A. We show that, in some sense, the commutative algebra O(A), in (1.1.2), is a limit of the algebras C[Repd A]GLd as d goes to infinity. A precise version of this statement will be proved in §4. The statement implies, in particular, that the Hilbert series of O(A) may be obtained as a limit of Hilbert series of the algebras C[Repd A]GLd as d goes to infinity. Observe further that the presentation A = F/J makes Repd A a closed subscheme in Repd F , the latter being a vector space. We introduce a noncommutative cyclic Koszul complex, K cyc q A to the GLd q A, such that each of the functors Repd , d ∈ ZI , sends K cyc fixed part of the standard Koszul complex of the subscheme Repd A ⊂ Repd F . Moreover, we prove that the two complexes become ‘asymptotically isomorphic’ as d → ∞. To be able to exploit usefully the ‘asymptotic isomorphisms’ above we introduce the notion of representation complete intersection (RCI) algebra, and a weaker (but more useful) notion of asymptotic RCI algebra. The algebra A = F/J is called RCI provided there exist arbitrarily large positive dimensions d such that the scheme Repd A is a complete